Double Marginalization in Performance-Based Advertising:
Implications and Solutions
Chrysanthos Dellarocas
Boston University
School of Management
dell@bu.edu
First draft: August 2009. Current draft: March 2010
Abstract
An important current trend in advertising is the replacement of traditional pay-per-exposure
(pay-per-impression) pricing models with performance-based mechanisms in which advertisers
pay only for measurable actions by consumers. Such pay-per-action (PPA) mechanisms are becoming the predominant method of selling advertising on the Internet. Well-known examples
include pay-per-click, pay-per-call and pay-per-sale. This work highlights an important, and hitherto unrecognized, side-effect of PPA advertising. I find that, if the prices of advertised goods are
endogenously determined by advertisers to maximize profits net of advertising expenses, PPA
mechanisms induce firms to distort the prices of their goods (usually upwards) relative to prices
that would maximize profits in settings where advertising is sold under pay-per-exposure methods. Upward price distortions reduce both consumer surplus and the joint publisher-advertiser
profit, leading to a net reduction in social welfare. They persist in current auction-based PPA
mechanisms, such as the ones used by Google and Yahoo. In the latter settings they always
reduce publisher revenues relative to pay-per-exposure methods. In extreme cases they also lead
to rat-race situations where, in their effort to outbid one another, advertisers raise the prices of
their products to the point where demand for them drops to zero. I show that these phenomena
constitute a form of double marginalization and discuss a number of enhancements to today’s
PPA mechanisms that restore equilibrium pricing of the advertised goods at efficient levels.
Keywords: performance-based advertising, sponsored search, keyword auctions, double marginalization, mechanism design
Acknowledgments: Many thanks to Jason Hartline, Lorin Hitt, De Liu, Michael Schwarz, Siva Viswanathan and
seminar participants at Harvard, Wharton, Tepper, Toulouse IDEI and Boston University for valuable feedback on
an earlier version of this work.
1
Half the money I spend on advertising is wasted;
the trouble is, I don’t know which half.
John Wanamaker, owner of America’s first department store
1
Introduction
John Wanamaker’s famous quote has been haunting the advertising industry for over a century. It
now serves as the motivation behind much of the innovation taking place in Internet-based advertising. From Google, Yahoo and Microsoft, to Silicon Valley upstarts, some of the best and brightest
technology firms are focusing a significant part of their energies on new mechanisms to reduce advertising waste. These come in many forms but have one thing in common: a desire to replace
traditional pay-per-exposure (also known as pay-per-impression) pricing models, in which advertisers pay a lump sum for the privilege of exposing an audience of uncertain size and interests to their
message, with performance-based mechanisms in which advertisers pay only for measurable actions
by consumers. Pay-per-click sponsored search, invented by Overture and turned into a multi-billion
dollar business by Google, Yahoo and other online advertising agencies, is perhaps the best known of
these approaches: advertisers bid in an online auction for the right to have their link displayed next
to the results for specific search terms and then pay only when a user actually clicks on that link,
indicating her likely intent to purchase. Pay-per-call, pioneered by firms such as Ingenio (acquired
by AT&T in 2007), is a similar concept: the advertiser pays only when she receives a phone call
from the customer, usually initiated through a web form. Pay-per-click and pay-per-call are viewed
by many as only an intermediate step towards what some in the industry consider to be the “holy
grail of advertising”: the pay-per-sale approach where the advertiser pays only when exposure to an
advertising message leads to an actual sale.1 All of these approaches are attempting to reduce all or
part of Wanamaker’s proverbial waste by tying advertising expenditures to consumer actions that
are directly related or, at least, correlated with the generation of sales. In the rest of the paper I
will refer to them collectively as pay-per-action (PPA) pricing models.
The current surge in pay-per-action advertising methods has generated considerable interest
from researchers in a variety of fields including economics, marketing, information systems and
computer science. This is important and timely since most of these methods have been invented
by practitioners and their properties and consequences are not yet fully understood. Although the
literature (surveyed in Section 2) has made significant advances in a number of areas, an important
area that, so far, has received almost no attention is the impact of various forms of PPA advertising
on the prices of the advertised products. With very few exceptions (also discussed in Section 2),
papers in this stream of research have made the assumption that the prices of the goods being
advertised are set exogenously and independently of the advertising payment method.
In this paper I make the assumption that the prices of the goods being advertised are an endogenous decision variable of firms bidding for advertising resources. I find that, if such prices
1
See, for example, "Pay per sale", Economist magazine, Sep. 29, 2005.
2
are endogenously determined by advertisers to maximize profits net of advertising expenses, PPA
advertising mechanisms induce firms to distort the prices of their goods (usually upwards) relative
to the prices that would maximize their profits in settings where there is no advertising or where
advertising is sold under pay-per-exposure methods. Upward price distortions reduce both consumer
surplus and the joint publisher-advertiser profit, leading to a net reduction in social welfare.
My results are driven by the fact that the publisher and the advertisers are independent profit
maximizing agents that interact in a “supply chain” type of relationship (i.e. the publisher supplies
an advertising slot to an advertiser who then uses it to advertise and sell a product to consumers).
When both of these agents independently set the prices they charge to their respective downstream
customers to optimize their profit margins, an inefficiency analogous to the well-known concept of
double marginalization occurs.
The phenomenon is very general and occurs in any setting where the publisher and advertisers
have some degree of market power. Specifically, I show that such price and revenue distortions also
arise in search advertising auctions, such as the ones currently used by Google and Yahoo. In the
latter settings they always lead to lower publisher revenues relative to pay-per-exposure methods.
Furthermore, in extreme cases where the set of competing advertisers are homogeneous with respect
to their valuation of an advertising slot, they lead to rat race situations where, in their effort to
outbid one another, advertisers raise the prices of their products to the point where demand drops
to zero.
I discuss how today’s auction-based search advertising mechanisms can be enhanced to restore
efficient pricing. I propose two broad sets of ideas. The first idea requires the advertiser to disclose
to the publisher both her bid as well as the product price she intends to charge. The publisher then
makes the advertiser’s current round quality weight a function of both her past history (which is
what the current generation of PPA keyword auctions already do) and her current round product
price (which is something current mechanisms don’t do). The second idea is to construct mechanisms whereby advertisers are charged some form of penalty, e.g. an auction participation fee,
following periods where no triggering actions (i.e. clicks, sales, etc.) occur. I show that the resulting
mechanisms maximize the joint publisher-advertiser profit and have identical revenue and allocation
properties to an optimal pay-per-exposure mechanism.
The rest of the paper is organized as follows. Section 2 discusses related work. Section 3
introduces the paper’s core intuitions through a simple posted price example setting. Section 4
illustrates methods of restoring efficient pricing in the context of our example setting. Section 5
shows how the ideas of the previous sections apply to the practically important case of auctionbased search advertising settings and examines their economic implications in that specific context.
Section 6 proposes ways of restoring efficient product pricing in auction-based search advertising.
Finally, Section 7 concludes.
3
2
Related Work
This work relates to a number of important streams of marketing, economics and computer science
literature. Nevertheless, the phenomenon discussed herein has so far not been addressed by any of
these literatures.
Sponsored search advertising
Pay-per-click online advertising, such as sponsored search links, is one of the most successful and
highly publicized methods of performance-based advertising. It is the main source of revenue for
sites like Google and Yahoo and one of the fastest growing sectors of the advertising industry. Not
surprisingly, this field has experienced an explosion of interest by both researchers and practitioners.
Important advances have been made on understanding the properties of the generalized second
price (GSP) auction mechanisms that are currently the prevalent method of allocating advertising
resources in such spaces (see, for example, Athey and Ellison 2008; Edelman et al. 2007; Varian
2007). A related stream of research has proposed several extensions to baseline GSP auctions that
aim to improve their properties. The following is an illustrative subset: Aggarwal et al. (2006)
propose an alternative advertising slot auction mechanism that is revenue-equivalent to GSP but
induces truthful bidding (GSP does not); Feng et al. (2007), Lahaie and Pennock (2007) and Liu
and Chen (2006) explore the allocative efficiency and publisher revenue implications of alternative
methods for ranking bidders, including “rank by bid” and “rank by revenue”; Katona and Sarvary
(2008) explore the equilibrium behavior of keyword auctions under more sophisticated assumptions
about users’ search behavior; Ashlagi et al. (2008) and Liu et al. (2008b) explore auction design in
the presence of competing publishers.
Growing attention is also being given to the perspective of advertisers bidding on such auctions;
the most important problems here include how to identify appropriate keywords (Abhishek and
Hosanagar 2007; Joshi and Motwani 2006; Rutz and Bucklin 2007) and how to dynamically allocate
one’s budget among such keywords (Borgs et al. 2007; Cary et al. 2007; Feldman et al. 2007;
Rusmevichientong et al. 2006). Finally, researchers have paid attention to incentive issues that are
inherent in pay-per-action advertising, most important among them being the potential for click
fraud, i.e. the situation where a third party maliciously clicks on an advertiser’s sponsored link
without any intention of purchasing her product (Immorlica et al. 2006; Wilbur and Zhu 2008) as
well as the advertiser’s incentive to misreport the frequency of her triggering action in order to avoid
paying the publisher (Agarwal et al. 2009; Nazeradeh et al. 2008).
The above theoretical and algorithmic contributions are complemented by a growing number of
empirical works (e.g. Animesh et al. 2009; Ghose and Yang 2009; Goldfarb and Tucker 2007; Rutz
and Bucklin 2008; Yao and Mela 2009b). For comprehensive overviews of current research and open
questions in sponsored search auctions the reader is referred to excellent chapters by Feldman et al.
(2008), Lahaie et al. (2007), Liu et al. (2008a) and Yao and Mela (2009a).
Interestingly, almost all papers on this burgeoning field assume that an advertiser’s value per sale
4
is exogenously given and do not consider how the performance-based nature of advertising affects the
advertiser’s pricing of the products being sold. The only two exceptions I am aware of are Chen and
He (2006) and Feng and Xie (2007). Chen and He (2006) study seller bidding strategies in a paidplacement position auction setting with endogenous prices and explicit consumer search. However,
they only assume a pay-per-exposure mechanism and derive results that are essentially identical to
my Proposition 7 i.e. (using the language of my paper) that advertisers price their product at the
point that maximizes their per-exposure value function. Feng and Xie (2007) focus on the quality
signaling aspects of advertising and propose a model that is in many ways orthogonal to mine. I
discuss their paper later in this section.
Performance-based contracting
Performance-based advertising is a special case of performance-based contracting. Contract theory
has devoted significant attention to such contracts, as they can help balance incentives in principalagent settings where moral hazard exists or where the sharing of risk between the two parties is
a concern (Holmstrom 1979; Holmstrom and Milgrom 1987, 1991). In the context of information
goods, Sundararajan (2004) studies optimal pricing under incomplete information about the buyers’
utility. He finds that the optimal pricing usually involves a combination of fixed-fee and usage-based
pricing. Closer to the context of this work, Hu (2006) and Zhao (2005) study how performancebased advertising contracts that optimally balance the incentives of both the publisher and the
advertise can be constructed. They both find that the optimal contract must have both a fixed and
a performance-based (i.e. PPA) component. Once again, however, both of these studies consider the
prices of advertised products as fixed and not as an endogenous decision variable under the control
of advertisers.
Advertising and Product Prices
The relationship between product prices and advertising has received quite a bit of attention in the
economics and marketing literature. These literatures have primarily focused on the quality signaling
role of prices in conjunction with advertising. The main result is that the simultaneous presence of
prices and advertising improves a firm’s ability to successfully signal its quality to consumers because
firms can partially substitute quality-revealing price distortions with quality-revealing advertising
expenditures (see, for example, Fluet and Garella 2002; Hertzendorf and Overgaard 2001; Milgrom
and Roberts 1986; Zhao 2000). Almost all works in this stream of literature assume that advertising
is sold under a traditional pay-per-exposure model.
The only exception I am aware of is Feng and Xie (2007). They study how the move from
exposure-based to performance-based advertising affects the ability of price and advertising to signal
product quality. Their main result is that such a move generally reduces the number of situations
where advertising expenditures can be used to signal quality and increases the prices charged to
consumers, since firms must now rely harder on the price signal to reveal their quality. Their result
relies on the assumption that higher quality firms are more likely to have a higher proportion of
5
repeat customers who would be clicking and purchasing the product anyway, but who nevertheless
induce incremental advertising charges in a pay-per-performance (PPP) model. Therefore, PPP
advertising is relatively more wasteful for high quality vs. low quality firms and this moderates a
high quality firm’s incentive to spend more on PPP advertising.
My results are orthogonal to this work since in my model price distortions are unrelated to the
advertisers’ desire to signal their quality or to the presence of repeat customers; they occur even
in settings where consumers have perfect knowledge of each advertiser’s quality or where repeat
customers do not exist. As I explain in Section 3, my results are driven by the fact that the
publisher and the advertisers are independent profit maximizing agents that interact in a “supply
chain” type of relationship (i.e. the publisher supplies an advertising slot to the advertiser who then
uses it to advertise and sell a product to consumers). When both of these agents independently set
the prices they charge to their respective downstream customers to optimize their profit margins,
an inefficiency akin to the well-known concept of double marginalization occurs.
In summary, the traditional literature on advertising has examined various aspects of the relationship between product prices and advertising expenditures in settings that essentially correspond
to what I call pay-per-exposure. On the other hand, the rapidly growing literature on performancebased advertising has largely assumed that product prices are exogenous to the choice of advertisement payment mechanism. This work breaks new ground by showing that when one endogenizes
product prices, performance-based advertising mechanisms create incentives for price distortions
that in most cases have negative consequences for most stakeholders.
3
The core intuitions
In this section I establish the connection between the double marginalization problem and price
distortions in performance-based advertising through the use of a simple posted price setting. I also
review some of the classic solutions to the problem.
3.1
The double marginalization problem
Consider a setting where an upstream monopolist manufacturer sells raw materials to a retailer who,
in turn, packages them and sells them to consumers. The retailer is a monopolist in the downstream
market. The manufacturer’s constant marginal production cost is denoted by cm and the retailer’s
marginal packaging cost by cr . The manufacturer sells the product to the retailer at wholesale price
w and the retailer sells the product to end consumers at a retail price p. (Figure 1a). Demand is
assumed to be linear, taking the form D(p) = d − p, where d is assumed to be greater than total
marginal cost c = cm + cr .
If the manufacturer and retailer are vertically integrated, the retail price p would be chosen
to maximize the total channel profit π(p) = D(p)(p − c). It follows that the profit maximizing
retail price equals
d+c
2 ,
yielding an efficient sales quantity of
d−c
2
and a channel profit of
(d−c)2
4 .
The manufacturing and retail departments can then negotiate a transfer price w to divide the total
6
Marginal
manufacturing
cost
Marginal
packaging
cost
Unit
retail
price
cr
cm
Unit
wholesale
price
p
Marginal
product
cost
Marginal
publisher’s
cost
cr
Unit
retail
price
p
w
w
Consumer
Retailer
Manufacturer
Consumer
Advertiser
Demand
D(p)=d-p
π r=D(p)(p-w-cr)
Retailer Profits
Manufacturer Profits
πm=D(p)(w-c )
Demand
D(p)=d-p
Advertiser Profits
π r=D(p)(p-w-c )
m
cm=0
(Per-sale)
publisher’s
fee
Publisher
r
Publisher Profits
πm=D(p)(w)
Joint Profits
Channel Profits
π r+ π m =D(p)(p-cr)
π r+ π m =D(p)(p-cr-cm)
(a) Classic supply chain setting
(b) Performance-based advertising setting
Figure 1: Analogies between a classic supply chain setting and performance-based advertising.
channel profit.
If the manufacturer and retailer are not vertically integrated, they independently choose their
respective prices to maximize their own profits. In this setting, the manufacturer moves first by
offering a wholesale price w. Consequently, the retailer faces a profit of πr (p|w) = D(p)(p − cr −
w) and responds by choosing a profit maximizing retail price
d+cr +w
.
2
Anticipating the retailer’s
reaction, the manufacturer chooses w to maximize his profit function πm (w|p) = D(p)(w − cm ) =
¡
¢
d − d+c2r +w (w − cm ). The profit maximizing wholesale price then equals d+cm2 −cr > cm (under
the assumption d > c = cm + cr ), inducing a retail price
level of
(d−c)2
16 .
d+c
2 .
3d+c
4
that exceeds the efficient retail price
Consequently, the manufacturer ends up with a profit of
The total profit realized in this channel shrinks to
3(d−c)2
16 ,
(d−c)2
8
and the retailer earns
which corresponds to only 75% of
the integrated channel profit. Consumers also end up clearly worse off since they pay a higher retail
price.
The above inefficiency stems from the double price distortion which occurs when two independent
firms (manufacturer, retailer) stack their price-cost margins, thus the term double marginalization.
The phenomenon has been identified by Cournot (1838) even thought its first concrete analysis is
most often attributed to Spengler (1950). It is easy to show that it occurs under any downward
sloping demand curve D(p) and that it is also not confined to monopoly settings; it is present in
any setting where the manufacturer and retailer each have some market power and therefore, mark
up their respective prices above marginal cost.
3.2
A survey of classic solutions
An extensive literature has proposed solutions to the double marginalization problem (see, for example, Jeuland and Shugan 1983; Rey and Verge 2008). Most solutions rely on the manufacturer
imposing some vertical restraints to the retailer. For instance, including a resale price maintenance
provision in a retail contract can directly stipulate the efficient retail price. Alternatively, a quantity
forcing contract specifies the minimum volume of sales the retailer must achieve. Setting this to the
integrated channel efficient quantity forces the retailer to choose the efficient price level. Both such
7
contracts involve the manufacturer directly interfering with the retailer’s pricing decision and are
often frowned upon, or legally unenforceable in some countries.
A less intrusive solution is to use a two part tariff. Specifically, the manufacturer charges the
retailer both a fixed franchise fee f and a marginal wholesale price w. The retailer profit function
becomes πr (p|w, f ) = D(p)(p−cr −w)−f . As before, for a given f the profit maximizing retail price
is
d+cr +w
2
with the corresponding retailer profit being
(d−cr −w)2
4
(d−cr −w)2
4
− f . The manufacturer can set the
− u where u is the reservation utility of the retailer. The manufacturer
¢
¡
2
thus chooses w to maximize πm (w|p, f ) = D(p)(w −cm )+f = d − d+c2r +w (w −cm )+ (d−cr4−w) −u.
franchise fee up to
The profit maximizing wholesale price then is w = cm , inducing the efficient retail price of p =
and an efficient integrated channel profit equal to
(d−c)2
4
d+c
2
. The central idea of two part tariffs in this
setting is to eliminate one of the margins: the manufacturer sets his price equal to marginal cost and
uses the franchise fee to gain its share of the surplus. As a result, the retailer faces a margin that
is identical to that of an integrated channel. With the price distortion removed channel efficiency is
restored.
The use of quantity discount contracts is yet another method for solving the double marginalization problem. For example, consider a contract that specifies a unit wholesale price equal to
w=
f
D
+ g, where f, g are non-negative constants and D is the total quantity bought by the retailer.
With such a scheme, the higher the consumer demand (and, therefore, the quantity ordered by the
retailer) the lower the retailer’s average cost. Since higher demand implies lower retail price, this
provides the retailer with incentives to keep her price low. In fact, it is easy to see that this price
contract is equivalent to a two-part tariff: for both contracts the total cost to the retailer for a given
quantity D is f + Dg. The quantity discount scheme can be viewed as an average-cost version of a
two-part tariff. Consequently, the reasoning outlined in the previous paragraph applies here as well.
If the manufacturer sets g = cm , the retailer’s equilibrium price will equal the retail price at the
efficient level. Then manufacturer and retailer can then divide the surplus by appropriately setting
f.
3.3
Analogy with performance-based advertising
In this section I show how the previous setting has a direct analogy to an advertising context.
Instead of a manufacturer, consider a monopolist publisher who owns an advertising resource, such
as a billboard located at a busy city square, a time slot in prime time TV, or space at the top of a
popular web page. Instead of a retailer, consider a monopolist advertiser who produces a product
or service at marginal cost cr . Since the advertising resource is an information good, I assume that
the publisher’s marginal cost cm is zero. I further assume that at least some consumers will not
be aware of the product unless the product is advertised on the publisher’s resource. Therefore,
gaining access to the publisher’s resource results in incremental demand for the product equal to
D(p) (Figure 1b).
Consider first a setting where access to the advertising resource is made using pay-per-exposure
(PPE) contracts. In such contracts the publisher charges the advertiser a flat fee f that is inde8
pendent of the incremental revenue that the advertiser is able to realize thanks to the advertising
resource. The advertiser’s profit function is then πr (p|f ) = D(p)(p − cr ) − f . It is easy to see that,
in such a setting, the publisher’s fee f does not impact the advertiser’s pricing problem. The advertiser will choose the price that maximizes her incremental sales profit. For example, if D(p) = d − p
(where d > cr ) the retail price would be set to
a channel profit of
(d−cr
4
)2
d+cr
2 ,
yielding an efficient sales quantity of
d−cr
2
and
that is split between the advertiser and the publisher through the use of
the flat fee f .
Let us now see how the situation changes when the publisher sells access to the advertising
resource using performance-based contracts. The most straightforward example of such contracts
is a pay-per-sale (PPS) contract whereby the advertiser pays the publisher a fee w for every sale
realized. In this setting the advertiser’s profit function is equal to πr (p|w) = D(p)(p−cr −w) and the
publisher’s profit function equal to πm (w|p) = D(p)w. The reader can readily verify that the above
profit functions are identical to those of the classic manufacturer-retailer double marginalization
example of Section 3.1 when cm = 0. With reference to that example, if D(p) = d − p, at equilibrium
the publisher will set his per-sale fee w at
d−cr
2
and the advertiser will price her product at
Notice that this price is higher than the sales profit maximizing price
d+cr
2
3d+cr
4 .
that the advertiser would
choose for her product if advertising was sold using a PPE contract. Accordingly, it leads to lower
demand. The publisher ends up with a profit of
profit realized in this channel shrinks to
(d−cr )2
8
3(d−cr )2
,
16
and the advertiser earns
(d−cr )2
16 .
The total
which corresponds to only 75% of the channel
profit if advertising was sold using PPE contracts. Consumers also end up clearly worse off since
they pay a higher retail price.
As is the case with the standard double marginalization problem, the above phenomena are fairly
general: they occur for any downward sloping demand curve D(p) and are also present in any setting
where the publisher and advertiser each have some market power and are, therefore, able to mark up
their respective prices above marginal cost. Finally, they are orthogonal to the specific mechanism
used by the publisher to allocate his resource to advertiser. More concretely, Section 5 shows that
similar price distortions are present in auction-based allocation methods, such as the ones used by
search engines to allocate online advertising space.
3.4
Generalization to arbitrary pay-per-action contracts
Pay-per-sale is only one example of a more general class of performance-based advertising contracts
that are often referred to by the term pay-per-action. This section shows how the phenomena
identified in the previous sections generalize to this broader class of contracts.
Pay-per-action (PPA) approaches make payment to the publisher contingent on a triggering
action that is either a sale, or some other consumer action (e.g. click, call) that has the following
properties:
1. It is linked to the advertising, i.e. only consumers who have been exposed to this particular
advertising message can perform the triggering action.
9
2. It is a necessary step of a consumer’s purchase decision process. This means that even though
not all consumers who perform the triggering action may buy the product, consumers cannot
purchase the product without performing the triggering action.
Let V (p) denote the ex-ante value the advertiser expects to obtain from leasing the advertising
resource. In most real-life settings this value will be equal to the incremental sales profit that the
advertiser expects to realize by leasing the resource and can thus be expressed as V (p) = D(p)(p−cr )
where p is the unit price of the advertised product, cr is the corresponding unit cost and D(p) is
the increase in demand due to advertising. This profit will be split between the advertiser and the
publisher and will, thus, also be referred to as the joint advertiser-publisher profit. Irrespective
of the precise functional form of V (p), assumptions 1-2 above allow us to uniquely express it as a
product V (p) = U (p)W (p) where:
U (p)
is the expected triggering action frequency (TAF) and
W (p)
is the expected value-per-action (VPA).
The precise meanings of U (p) and W (p) depend on the specifics of the payment contract. For
example:
• In pay-per-sale contracts U (p) is equal to the incremental demand due to advertising while
W (p) is equal to the unit profit.
• In pay-per-click contracts U (p) is equal to the per-period audience size times the probability
of a click (the clickthrough rate) and W (p) is equal to the conditional probability of a purchase
given a click (the conversion rate) times the unit profit.
• Traditional per-per-exposure contracts are a special case of the above framework where U (p) =
1 and W (p) = V (p).
A special case of particular importance in online settings is that of personalized ads, such as the
ones displayed in response to a keyword search. In such settings the size of the audience is one and
function U (p) denotes the triggering action probability.
In the rest of the paper I am assuming that the publisher has a reliable way of obtaining correct
estimates of U (p) and, therefore, that strategic misreporting of triggering actions from the part of
the advertiser is not an issue.2
Consider now a general pay-per-action (PPA) contract whereby the advertiser pays the publisher
a fee w every time a triggering action occurs. In this more general setting the advertiser’s profit
function is equal to πr (p|w) = U (p)(W (p) − w) = V (p) − U (p)w and the publisher’s profit function
equal to πm (w|p) = U (p)w. Let p∗ be the price that maximizes V (p). It is easy to see that this
is the price that maximizes πr (p|w) when w = 0. It is also (see Section 3.3) equal to the product
2
Addressing an advertiser’s incentive to misreport the frequency of payment triggering action to the publisher is
an important consideration in pay-per-action schemes but orthogonal to the focus of this paper. See Agarwal et al.
(2009) and Nazerzadeh et al. (2008) for discussion and proposed solutions.
10
price pE that the advertiser would choose under a pay-per-exposure contract. Now, in any PPA
setting where the publisher has some market power, he will charge a price w > 0 (otherwise, he will
make no profits). From monotone comparative statics theory (Topkis 1998) it follows that if the
cross-partial derivative
∂πr2
∂w∂p
is non-negative (non-positive) everywhere, the profit maximizing price
pA = arg maxp πr (p|w) is a monotonically increasing (decreasing) function of w. It is
∂πr2
∂w∂p
= −U 0 (p),
which is non-negative if and only if U 0 (p) ≤ 0 for all p. Under these conditions, if w > 0 it will be
pA > p∗ = pE , leading to joint advertiser-publisher profit V (pA ) < V (p∗ ) = V (pE ). I have thus
shown the following result:
Proposition 1: In all PPA settings where (i) both the publisher and the advertiser have some
market power and (ii) the triggering action frequency satisfies U 0 (p) ≤ 0 for all p, with the inequality
strict for at least some p, the following hold:
1. The advertiser prices her products at a price pA that is strictly higher than the price pE she
would charge if advertising was sold using pay-per-exposure.
2. The above price distortion leads to lower consumer surplus and a lower joint advertiserpublisher profit compared with a setting where advertising was sold using pay-per-exposure.
The assumption U 0 (p) < 0 implies that at least some consumers have some (possibly imperfect)
information about the price of the products being advertised.3 This assumption obviously holds in
the context of pay-per-sale contracts. It also holds in pay-per-click or pay-per-call settings where
either (i) product prices are displayed on the ad, or (ii) a section of the consumer population has
access to such prices before clicking or calling through separate information channels (e.g. product
reviews, price comparison charts, previous clicks or calls to the same company, etc.) or (iii) a section
of the population has sufficient industry knowledge (i.e. “knows” something about the advertiser’s
value function and the publisher’s price structure) to be able to infer (possibly imperfectly) these
prices.
4
Restoring efficiency in performance-based advertising
In Section 3.2 I discussed two classic solutions to the double marginalization problem: two part tariffs
and quantity discounts. I begin this section by arguing that none of these solutions satisfactorily
applies to performance-based advertising. The rest of the section then proposes two new ideas that
can be used to eliminate price distortions in such settings.
The equivalent of a two part tariff in the performance-based advertising setting would be a
contract where the publisher charges the advertiser a fixed fee f and a per-action fee equal to his
marginal cost. The assumption of zero publisher marginal cost implies that there would only be a
3
If no consumer knows anything about the price of an advertised product before performing the action (click, call,
sale, etc.) that triggers payment to the publisher then it will be U 0 (p) = 0 for all p. The advertiser’s action of raising
the price of her product then has no effect in terms of reducing the expected total payment to the publisher. In such
a setting the advertiser has no incentive to deviate from the price p∗ that maximizes the value she obtains from the
advertising resource.
11
fixed fee. But this would be equivalent to a pure PPE contract. Two part tariffs, are thus, not useful
in restoring efficiency if one wants to retain a performance-based compensation component. For zero
marginal cost, quantity discount contracts translate to contracts where the per-action fee charged by
the publisher would be equal to
f
D,
i.e. inversely proportional to the volume D of incremental sales
due to advertising (or, more broadly, equal to
f
U,
i.e. inversely proportional to the triggering action
frequency U ). The problem here is that sales (or, more broadly, triggering actions) only materialize
after the advertiser obtains the resource and, sometimes, are spread out over a substantial time
period. Effective quantity discount contracts are often impractical in such cases. The problem is
even more severe in personalized ad settings where the triggering action occurs either once (in which
case the advertiser pays the publisher) or not at all (in which case the advertiser pays nothing). In
such cases quantity discount contracts are infeasible.
Although the standard approaches to eliminating inefficiencies due to double marginalization
do not readily apply to performance-based advertising, it is possible to construct variants of these
approaches that are better suited to this context. The rest of this section proposes two such ideas.
4.1
Price-based fee schedules
The main complication of applying the quantity discount idea in the context of performance-based
advertising is that the incremental quantity sold because of advertising (or, more broadly, the triggering action frequency) is only known ex-post. However, if the publisher has sufficiently precise
knowledge of the advertiser’s expected triggering action frequency function U (p) and if the advertiser
can credibly disclose her product price p to the publisher, the publisher can construct a contract
where the per-action fee charged to the advertiser can be set ex-ante equal to
f
U (p) ,
i.e. inversely
proportional to the expected triggering action frequency given the advertiser’s price. It is then
straightforward to prove the following result:
Proposition 2. In PPA settings where the advertiser discloses her product price p and the
publisher sets the per-action fee charged to the advertiser equal to
f
U (p) ,
where f is a constant chosen
to make the advertiser’s participation individually rational and U (p) is an accurate assessment of
the advertiser’s expected triggering action frequency (or probability) as a function of product price,
the advertiser prices her products at the point pA = p∗ that maximizes the joint advertiser-publisher
profit.
The above solution is effective in a wide range of settings. It works well in settings where the
triggering action frequency is a stochastic function of product price and also in the important special
case of personalized ad settings where U (p) is a probability function. On the minus side it requires
the advertiser to ex-ante commit to a product price and the publisher to be able to monitor the
extent to which the advertiser adheres to her commitment.
4.2
History-based fee schedules
In settings where the advertiser and the publisher interact repeatedly, the publisher can use the
“shadow of the future” to induce the advertiser to price her products at the joint profit-maximizing
12
level. The applicable solutions differ depending on whether the triggering action frequency is a
deterministic or stochastic function of product price.
Deterministic settings.
In deterministic settings the publisher can induce efficient product
pricing by committing to make next period’s per-action fee a declining function of the current
period’s observed triggering action frequency: the more often the advertiser pays the publisher
today, the lower the per-action fee she will have to pay tomorrow. This scheme gives the advertiser
a disincentive to increase her current period product price. If U 0 (p) ≤ 0, as product price increases,
the current period’s triggering action frequency declines. A lower triggering action frequency leads
to a lower total advertising expenditure this period but (because of the increase in next period’s
per-action fee) to a higher advertising expenditure next period.
The following proposition shows that, in settings where the triggering action frequency is a
deterministic function of product price, the publisher can always design a per-action fee schedule
that balances these two forces and induces the advertiser to price her products at the point that
maximizes the joint advertiser-publisher profit.
Proposition 3. Consider an infinite horizon repeated PPA setting where (i) the triggering action
frequency is a deterministic function U (p) of product price and (ii) the per-action fee wt (t = 0, 1, ...)
charged by the publisher to the advertiser at period t has the following expression:

 w0
h
³
wt =
 w0 1 + 1 1 −
δ
Ut−1
U (p∗ )
´i
if t = 0
if t > 0
where 0 < w0 < W (p∗ ), p∗ = arg maxp V (p), U (p) is the triggering action frequency function as a
function of product price p, Ut−1 is the observed triggering action frequency at period t − 1 and δ is
the advertiser’s discount factor.
∗
In such a setting for all t = 0, 1, ... the advertiser will price her products at the level pA
t = p that
maximizes the joint advertiser-publisher profit.
Stochastic settings. If the relationship between product price and the observed triggering
action frequency (or probability) is stochastic, repeated interaction can still induce efficient pricing.
However, the solution now always involves the publisher charging advertisers some fixed (i.e. nonperformance-based) fees in addition to action-based fees, at least some of the time.
I will convey the intuition behind these ideas by restricting my attention to personalized ad
settings, i.e. settings where the audience size is equal to one and the triggering action probability
is U (p). Such settings are practically important in online domains as well as analytically tractable.
According to Proposition 2, the publisher can induce the advertiser to price her product at the
point p∗ that maximizes the joint advertiser-publisher profit by asking her to disclose her current
period product price pA and then setting the current period per-action fee equal to
f
.
U (pA )
If the
advertiser’s (per-period) reservation utility is u, the publisher can set the constant f as high as
V (p∗ ) − u, resulting in discounted lifetime publisher profits
13
1
1−δ
[V (p∗ ) − u] and discounted lifetime
advertiser profits
u
1−δ .
If the publisher cannot credibly learn the advertiser’s current period product price he can still
induce her to price at point p∗ by setting up a schedule of fixed and per-action fees that depend
on the advertiser’s history of past triggering actions. For example, I will prove that the following
two-state mechanism can achieve the desired objective:
Mechanism M1:
State H:
All advertisers begin the game at state H. While they are in state H advertisers are
charged a per-action fee wH . If a triggering action occurrs during the current period the
advertiser remains in state H, otherwise she transitions to state L.
State L:
For every period that advertisers are in state L they are charged a fixed fee f and a
per-action fee wL . If a triggering action occurrs during the current period the advertiser
transitions to state H otherwise she remains in state L.
The following result provides the details:
Proposition 4. Consider an infinite horizon PPA setting that is characterized by joint advertiserpublisher profit V (p), triggering action probability U (p) and the above two-state mechanism M1. The
advertiser’s per-period reservation utility is u. If the publisher sets:
f = (V (p∗ ) − u) 1δ
wH = wL = (V (p∗ ) − u)
the advertiser always prices her products at pA = p∗ and the publisher extracts the maximum lifetime
profits that are consistent with the advertiser’s reservation utility.
The gist of the above idea is to give the advertiser an incentive to increase the probability of a
triggering action (by lowering her price) by charging her an additional fixed fee on periods that immediately follow periods where the advertiser leased the resource but there was no triggering action.
Observe that, compared to a pure PPA solution (e.g. the price-based fee schedule of Section 4.1),
this solution involves a lower per-action fee (V (p∗ ) − u versus
fixed fee (V
(p∗ )
− u)
1
δ.
V (p∗ )−u
U (p∗ ) )
coupled with an occasional
The discounted payoffs of both the advertiser and the publisher are identical
to those obtained using a pure PPA (e.g. price-based fee) solution.
If advertisers are averse to the idea of paying a non-performance-based fee to the publisher,
it is possible to design a variant of this approach where the expected present value of the nonperformance-based fees is equal to zero. The idea is to extend Mechanism M1 so that, in addition
to charging advertisers a fixed fee f when they are at state L, the publisher also offers them a fixed
subsidy s when they are at state H such that the discounted lifetime sum of the fees and subsidies
is zero. The following result provides the details:
Proposition 5. Consider an infinite horizon PPA setting that is characterized by joint advertiserpublisher profit V (p), triggering action probability U (p) and an extension of mechanism M1 that in14
volves a subsidy s while at state H and a fee f while at state L. The advertiser’s per-period reservation
utility is u. If the publisher sets:
∗
(p )
s = (V (p∗ ) − u) 1−U
U (p∗ )
∗ ))
(p
f = (V (p∗ ) − u) 1−δ(1−U
δU (p∗ )
wH = wL =
V (p∗ )−u
U (p∗ )
then (i) the advertiser always prices her products at pA = p∗ , (ii) the discounted lifetime sum of
the non-performance-based fees and subsidies is equal to zero and (iii) the publisher extracts the
maximum lifetime profits that are consistent with the advertiser’s reservation utility.
Notice that in the above result the per-action fee is equal to the per-action fee
V (p∗ )−u
U (p∗ )
in a pure
PPA setting. This is intuitive since the system of non-performance-based fees and subsidies has zero
net present value so the publisher obtains his net revenue from the per-action fees only.
All preceding solutions require the introduction of non-performance-based fees in addition to
performance-based fees, at least part of the time. It is tempting to think that one could perhaps
design a mechanism that induces optimal pricing using performance-based fees alone. The idea would
be to charge a higher per-action fee following a period where there was no triggering action and a
lower fee following a period where such action occurred. As was the case above, such mechanisms
will give the advertiser additional incentives to increase the probabiluty of a triggering action by
keeping her price low. The following result shows that such schemes are indeed possible to design.
However all such mechanisms where the publisher charges positive per-action fees must involve a
rule whereby the advertiser is expelled from the game after k consecutive periods where no triggering
action occurs. Furthermore, they can never be as efficient as mechanisms that charge both fixed and
per-action fees.
Proposition 6: All pure PPA mechanisms that induce advertisers to always price their products
at the point pA = p∗ that maximizes the per-period joint advertiser-publisher profit V (p) have the
following properties:
1. Either the publisher’s per-action fees are zero for all periods or advertisers are expelled from
the game after k consecutive periods where no triggering action occurs, for some integer k ≥ 1.
2. In all such mechanisms the advertiser’s discounted lifetime payoff (net of publisher fees) is
equal to
1−δ k
1−δ V
(p∗ ).
3. For all finite k, the discounted lifetime joint publisher-advertiser profit is strictly lower than
what can be achieved by an infinite horizon two-state fee schedule that involves both fixed and
per-action fees (such as mechanism M1).
The detailed expressions of the per-action fees imposed by the publisher to the advertiser and of the
associated publisher profits as a function of k can be obtained for special cases of such mechanisms.4
4
Appendix A provides a full analysis of such a special case and also shows that there exists an optimal k that
maximizes the publisher’s discounted lifetime profits.
15
The intuition behind Part 3 of the above result is simple. The probability of k consecutive periods
with no triggering action (1 − U (p∗ ))k is always positive (though, for large k, potentially very small).
Therefore, with virtual certainty, all advertisers will be expelled from the game after a sufficiently
large finite number of periods. For that reason, the social efficiency of this mechanism can never be
as high as that of a game where the advertiser is never expelled. In practice, however, it can get
arbitrarily close.
A note about incomplete information settings. The analysis of this Section assumes that
the publisher has perfect knowledge of the advertiser’s payoff function. In real-life applications this is
seldom the case. However, if there is repeated interaction between the advertiser and the publisher,
as it is common in real-life advertising settings, the literature on strategic learning (Young 2004;
Young 2007) has shown that under a wide range of assumptions it is possible for the publisher to
learn the advertiser’s payoff function within an arbitrarily small margin of error. The results of this
paper then hold at the limit where the publisher has fully learned the advertiser’s payoff function.
5
Double marginalization in search advertising auctions
This Section shows that the price distortions that were identified in Section 3 occur in the practically
important case of sponsored search keyword auctions. It also analyzes the economics implications
of these distortions for the various stakeholders.
Consider, as before, a setting where a monopolist publisher owns an advertising resource and
leases it on a per-period basis to a heterogeneous population of N advertisers. Advertisers are
characterized by a privately known unidimensional type q ∈ [q, q], independently drawn from a
distribution with CDF F (q). An advertiser’s type relates to the attractiveness of her products or
services to consumers; I assume that ceteris paribus higher types are, on average, more attractive.
In the rest of the paper I will refer to q as the advertiser’s quality, even though other interpretations
are possible.5 An advertiser’s quality affects the ex-ante value V (p, q) she expects to obtain from
leasing the advertising resource for one period. As in Section 3.4 value V (p, q) will be equal to the
incremental sales profit that the advertiser expects to realize by leasing the resource.
In the rest of the paper I will use the notation fi (x1 , ..., xn ) , i = 1, ..., n to denote the partial
derivative of f (·) with respect to its ith variable. For example, V2 (p, q) =
assumed to hold for all p ∈
R+
∂V (p,q)
∂q .
The following are
and q ∈ [q, q]:
A1
V (p, q) is unimodal in p, attaining its unique maximum at some p∗ (q) > 0
A2
limp→∞ V (p, q) = 0
A3
V2 (p, q) > 0
5
For example, in settings with network effects (e.g. when the advertisers are social networks) q can be the size of
the advertiser’s user base.
16
A1 and A2 are common and intuitive consequence of treating V (p, q) as a sales profit function. A3
implies that some information about an advertiser’s type becomes available to consumers at some
point during the advertising-purchasing process, but still allows for a fairly general range of settings
(for example settings where this information might be noisy, where only a subset of consumers are
informed, where firms might attempt to obfuscate their true types, etc.).
Each period the publisher allocates the resource to one of the competing firms using a Vickrey
auction. Auction-based allocation of advertising resources is the norm in sponsored search advertising and is also not uncommon in offline settings (e.g. superbowl ads). The double marginalization
effects I discuss in this paper are orthogonal to whether the publisher offers one or several (identical
or vertically-differentiated) resources. This allows us to ignore the multi-unit mechanism design
complications present, say, in sponsored search position auctions (Athey and Ellison 2008; Edelman
et al. 2006; Varian 2007) and focus on a single-unit auction.
Traditional pay-per-exposure (PPE) methods charge advertisers a fee that is levied upfront and
is independent of the ex-post value that advertisers obtain by leasing the resource. In a Vickrey
auction setting each advertiser bids the highest such fee she is willing to pay. The auctioneer allocates
the resource to the top bidder and charges her the second-highest bid. Assuming that every other
£ ¤0
bidder of type y bids an amount equal to β E (y) and that, as I will later show, it is β E (y) ≥ 0,
at a symmetric Bayes-Nash equilibrium an advertiser of type q bids bE (q) and sets the price of her
product at pE (q) to maximize her net expected profit:
−1
ΠE (q; bE (q), pE (q), β E (·)) =
[β E ] ˆ (bE (q))
¡
¢
V (pE (q), q) − β E (y) G0 (y)dy
(1)
q
where G(y) = F N −1 (y) is the probability that the second highest bidder’s type is less than or equal to
y and G0 (y) is the corresponding density.6 At equilibrium it must also be β E (q) = bE (q). The above
specification subsumes the special case where product prices p(q) are given exogenously. In the latter
case, a bidder of type q simply chooses a bid bE (q; p(·)) that maximizes ΠE (q; bE (q; p(·)), p(q), β E (·))
subject to β E (q) = bE (q; p(·)).
I use the following shorthand notation:
ΠE (q)
advertisers’ PPE equilibrium profit function (endogenous product prices)
ΠE (q; p(·))
advertisers’ PPE equilibrium profit function (exogenous product prices)
According to standard auction theory (e.g. Riley and Samuelson 1981) the expected publisher
revenue associated with bids β(y) is equal to:
6
Throughout this paper I restrict my attention to symmetric Bayes-Nash equilibria. Unless specified otherwise,
all subsequent references to “equilibrium” thus imply “symmetric Bayes-Nash equilibrium”.
17
ˆq
RE (β(·)) = N
q


ˆz

 0
0
 β(y)G (y)dy  F (z)dz
(2)
q
I use the following shorthand notation:
RE = RE (bE (·))
publisher’s PPE equilibrium revenue (endogenous product prices)
RE (p(·))
publisher’s PPE equilibrium revenue (exogenous product prices)
=
RE (bE (·; p(·)))
The following result characterizes the most important properties of allocating the resource using
PPE bidding
Proposition 7: In PPE settings where advertisers endogenously set the prices of their advertised
products to maximize profits net of advertising expenditures:
1. Advertisers bid their expected ex-ante value of acquiring the resource, given their price:
bE (q) = V (pE (q), q)
2. Advertisers set the price of their products at the point that maximizes their ex-ante expected
value of acquiring the resource:
pE (q) = arg max V (p, q) = p∗ (q)
p
3. Equilibrium PPE publisher revenues are equal to or higher to publisher revenues obtained in
any PPE setting where the products of advertised products are set exogenously:
RE = RE (p∗ (·)) ≥ RE (p(·))
with the inequality strict if and only if p∗ (q) 6= p(q) for at least one q ∈ [q, q].
The principal takeaway of the above proposition is that auction-based PPE mechanisms induce
all competing advertisers to price their products at the point that maximizes the joint advertiserpublisher profit. This is the point that maximizes the social efficiency, as well as the publisher
revenues that are attainable through the use of the auction mechanism.
Pay-per-action (PPA) approaches make payment to the publisher contingent on a triggering
action that is either a sale, or some other consumer action (e.g. click, call) that (i) is linked to the
advertising, i.e. only consumers who have been exposed to this particular advertising message can
perform the triggering action and, (ii) is a necessary step of a consumer’s purchase decision process.
As discussed in Section 3.4 the above assumptions allow us to uniquely express the advertiser’s exante value function as a product V (p, q) = U (p, q)W (p, q) where U (p, q) is the expected triggering
action frequency (or, in the special case of personalized ads, the triggering action probability) and
W (p, q) is the expected value-per-action.
18
Early PPA sponsored search mechanisms used a simple rank-by-bid (RBB) allocation rule: Advertisers bid the maximum per-action fee they were willing to pay. The slot was allocated to the
highest bidder who paid a per-action fee equal to the second highest bid. RBB mechanisms do not
always allocate the resource to the advertiser that values it the most and, thus, do not optimize
publisher revenues.7 For that reason they were quickly abandoned in favor of, more sophisticated,
rank-by-revenue mechanisms. The idea behind rank-by-revenue (RBR) is the following: The publisher computes a quality weight ui for each advertiser. The quality weight is typically based on
past performance data and attempts to approximate that advertiser’s expected triggering action
frequency U (p, q). Once bidders submit their bids bi the publisher computes a score si = ui bi for
each bidder. Assuming that (as we will show) advertisers place bids equal to their expected valueper-action W (p, q), the score is an estimation of the expected total value V (p, q) = U (p, q)W (p, q)
that the advertiser will generate by acquiring the resource. The publisher allocates the resource to
the bidder with the highest score and charges the winning bidder an amount u2 b2 /u1 equal to the
second highest score divided by the winning bidder’s quality weight. It has been shown that RBR
methods have better allocative efficiency and auctioneer revenue properties than RBB (Feng et al.
2007; Lahaie and Pennock 2007; Liu and Chen 2006). Both Google and Yahoo use variants of this
mechanism in their sponsored link auctions. For these reasons, the rest of the section will focus on
RBR PPA mechanisms.
Let Φ(q, s) denote an advertiser’s beliefs about every other bidder’s joint quality (q) and score (s)
distribution. At equilibrium these beliefs must be consistent with bidding and publisher behavior.
Let Φ(q) ≡ F (q) and Φ(s) be the corresponding marginal distributions and let Ψ(s) = [Φ(s)]N −1
denote the advertiser’s belief that every other bidder’s score will be less than s . Denote the
advertiser’s current quality weight as u. The single period specification of the advertiser’s RBRPPA bidding problem is to choose a bid bA (q, u) and a price pA (q, u) that maximize:
A (q,u)
ubˆ
³
s´ 0
Ψ (s)ds
U (pA (q, u), q) W (pA (q, u), q) −
u
ΠA (q, u; bA (q, u), pA (q, u)) =
0
(3)
The corresponding single period RBR-PPA publisher revenue is equal to:
ˆq
RA (bA (·, ·), pA (·, ·)) = N




Eu|q U (pA (q, u), q) 
q
A (q,u)
ubˆ
0

s 0

Ψ (s)ds F 0 (q)dq
u
(4)
where Eu|q [·] denotes expectation with respect to u conditional on an advertiser’s type being q.
The publisher’s objective is to use u as an approximation of an advertiser’s triggering action
frequency. Of particular interest, therefore, is the behavior of the system at the limit where the
7
For example, consider a setting where the per-action value W (p, q) and the total value V (p, q) = U (p, q)W (p, q)
are negatively correlated for all q. Then, the advertiser who has the highest per-action value W (p, q) will be able
to place the highest bid and win the resource, even thought the total value she expects to generate by leasing the
resource is the lowest among all advertisers.
19
publisher has obtained “correct” estimates of all quality weights, i.e. where each quality weight is
equal to the respective advertiser’s equilibrium triggering action frequency:
ui = U (pA (qi , ui ), qi )
I use the following shorthand notation to refer to equilibrium quantities in such “correct quality
weight” equilibria:
pA (q) = pA (q, U (pA (q), q))
equilibrium product prices
bA (q)
equilibrium bids
=
bA (q, U (pA (q), q))
ΠA (q)
=
RA
RA (bA (q), pA (q))
=
ΠA (q, U (pA (q), q); bA (q), pA (q))
equilibrium advertiser’s profits
equilibrium publisher’s revenue
The following proposition summarizes equilibrium bidding behavior and revenues in the above
setting:
Proposition 8: If advertising is sold on an RBR PPA basis, the following hold:
1. Advertisers set the price of their products at a point pA (q, u) that has the following properties:
pA (q, u) > p∗ (q)
if U1 (p, q) < 0 for all p
pA (q, u)
<
p∗ (q)
if U1 (p, q) > 0 for all p
=
p∗ (q)
if U1 (p, q) = 0 for all p
pA (q, u)
2. In settings that admit interior solutions advertisers bid their expected ex-ante value per action
given their price:
bA (q, u) = W (pA (q, u), q)
3. In settings that admit interior solutions and where, additionally (i) the publisher sets every
advertiser’s quality weight to her respective equilibrium triggering action frequency and (ii)
equilibrium value functions V (pA (q), q) are monotonically increasing in q, product prices pA (q)
are solutions of the following equation:
V1 (pA (q), q)G(q) − U1 (pA (q), q)Z A (q) = 0
where Z A (q) =
³´
q
q
(5)
´
V (pA (y), y)G0 (y)dy /U (pA (q), q) is the expected per-action payment to
the publisher.
Proposition 8 shows that the price distortion due to the double marginalization effect that is induced
by performance-based advertising persists in RBR PPA settings. The rest of the section will show
that, in the special context of RBR-PPA auctions this price distortion has a series of negative implications: It always reduces consumer surplus, social welfare and publisher revenues. Furthermore,
in settings where the competing advertisers are homogeneous it induces a rat race situation where
20
all advertisers raise their prices at the point where demand approaches zero. Just as I did in Section
3.4, I will focus my attention on the practically important case U1 (p, q) < 0.
Impact on consumers and social welfare
From Proposition 8, if U1 (p, q) < 0 it is pA (q) > p∗ (q) for all q: product prices increase, reducing
consumer surplus. Furthermore, the joint advertiser-publisher profit V (pA (q), q) is then always less
than the optimum V (p∗ (q), q). If we define social welfare as the sum of consumer surplus plus the
joint advertiser-publisher profit the following corollary immediately ensues:
Corollary 1: Consumer surplus and social welfare are strictly lower in a RBR-PPA mechanism
with endogenous product prices and perfectly estimated quality weights than in a corresponding PPE
mechanism.
Impact on publisher revenues
The key result here is that, if every advertiser’s quality weight is a correct assessment of her equilibrium triggering action frequency, RBR-PPA is allocation and revenue equivalent to a PPE setting
where product prices are exogenously set to pA (·).
Proposition 9: If advertising is sold on a RBR-PPA basis and the publisher sets every advertiser’s quality weight to her respective equilibrium triggering action frequency then:
1. Advertiser profits are identical to her equilibrium profits in a PPE setting where prices are
exogenously set to pA (·):
ΠA (q) = ΠE (q; pA (·))
2. Publisher revenues are identical to his equilibrium revenues in a PPE setting where every
advertiser exogenously prices her products at pA (q):
RA = RE (pA (·))
Once we have this result in place it is easy to show that RBR-PPA results in lower publisher revenues
than traditional PPE. By Proposition 8, in general it will be pA (·) 6= p∗ (·). From Proposition 7 (Part
3) it will then be RE = RE (p∗ (·)) > RE (pA (·)) = RA . Therefore:
Corollary 2: Equilibrium publisher revenues are strictly lower in a RBR-PPA mechanism with
endogenous product prices and perfectly estimated quality weights than in a corresponding PPE mechanism.
Rat race phenomena
I will now show that, for given U (p, q) and W (p, q), the magnitude of the price distortion induced by
a RBR-PPA mechanism has a positive relationship with the ratio of a bidder’s expected (per-action)
payment relative to her value-per-action. When this ratio tends to one, the advertiser tends to raise
21
the price of her products to the point where demand drops to zero and the resource becomes useless.
This happens when the set of advertisers competing for the resource are homogeneous with respect
to the value they expect to generate by leasing the advertising resource.
Proposition 10. Let:
Z A (q)
ζ (q) =
=
W (pA (q), q)G(q)
A
´q
q
V (pA (y), y)G0 (y)dy
V (pA (q), q)G(q)
(6)
denote the expected (per-action) payment-to-valuation ratio of an advertiser of type q conditional on
that advertiser winning the publisher’s auction. Fixing U (p, q) and W (p, q), the following statements
summarize how the magnitude of ζ A (q) impacts equilibrium RBR-PPA prices and the equilibrium
value of the advertising resource to the winning advertiser:
1.
∂pA (q;ζ A (q))
∂ζ A (q)
≥0
2. If W1 (p, q) > 0 for all p then it is limζ A (q)→1 V (pA (q; ζ A (q)), q) = 0
The intuition behind this result is the following: The higher the per-action payment to the publisher,
the higher the advertisers’ marginal gain from increasing pA (q) and thus reducing the triggering
action frequency (i.e. the frequency of paying the publisher). At the limit where the expected peraction payment Z A (q) approaches a bidder’s value-per-action W (pA (q), q) an advertiser’s losses from
the reduction in demand that results from price increases are almost exactly compensated by the
corresponding reduction in the payment to the publisher. At the same time, if W1 (p, q) > 0, higher
product prices result in a higher value-per-action, which allows the advertiser to place a higher bid.
Competition among bidders for the advertising resource then pushes product prices upwards to the
point where the triggering action frequency (and thus the value of the resource to the advertiser)
goes to zero. This is a rat-race situation that, clearly, has negative consequences for all parties
involved.
Integrating (6) by parts gives:
´q
ζ A (q) = 1 −
∂
q ∂y
£
¤
V (pA (y), y) G(y)dy
V (pA (q), q)G(q)
(7)
From (7) it follows that ζ A (q) is inversely related to the variability of the bidder population’s
equilibrium value V (pA (y), y) as a function of the bidders’ type. The more homogeneous the value
across bidders, the smaller the distance between the valuations of any two consecutive bidders and
thus the higher the expected payment relative to the winning bidder’s value. At the limit where
£
¤
∂
A (y), y) → 0, it is ζ A (q) → 1. Intuitively, if the bidder population is homogeneous with
V
(p
∂y
respect to the value it expects to generate from advertising, the bidding competition for obtaining
the resource becomes more intense and drives product prices up to the point where demand drops
to zero.
22
6
Restoring efficiency in search advertising auctions
Section 4 discusses several ways of restoring efficient pricing in a posted price performance-based
advertising setting. This section adapts some of these ideas to the context of search advertising
auctions. The key difference in this context is that the publisher does not have direct control of
per-action fees since these are decided by the auction mechanism.
6.1
Price-based quality weights
In Section 4.1 I discussed how the idea of using quantity discounts to induce efficient pricing in
settings that are prone to double marginalization can be adapted to the context of performancebased advertising: the publisher asks the advertiser to disclose her current round product price p and
charges a per-action fee that is inversely proportional to the advertiser’s expected triggering action
frequency U (p), given her price. This idea is not directly applicable in auction-based mechanisms
where the publisher cannot directly set per-action fees. However, in RBR schemes the publisher has
indirect control over both the probability that an advertiser will obtain the resource as well as over
the price that she is going to pay through her quality weight u. It is easy to show that:
Lemma 1: For all q, u it is
∂ΠA (q,u)
∂u
≥0
Recall that an advertiser’s quality weight is meant to be an estimate of her expected triggering
action frequency. RBR mechanisms, therefore, offer a “bonus” to advertisers that can increase their
triggering action frequency. In that sense, they have a form of “quantity discount” already built
into them. The problem, however, is that most current implementations of RBR-PPA treat each
advertiser’s quality weight as a point estimate that is based on the advertiser’s history on past
rounds only. Furthermore, once the estimation process converges the quality weight becomes fixed
and independent of the advertiser’s current period product price. Proposition 8 shows that price
distortions persist at that limit.
If the publisher has an accurate estimate of each advertiser’s entire triggering action frequency
function U (p, qi ) and if reliable elicitation of each round’s product prices is possible, the following
result shows that the publisher can induce efficient product pricing by making each advertiser’s
current round quality weight a function of both her past history and current period product price.
Proposition 11: Consider a RBR-PPA mechanism where:
1. Each round advertisers i = 1, ..., N are asked to disclose their bid bi and current round product
price pi
2. Each advertiser’s current round quality weight is set to ui = Ũi (pi )
At the limit where the publisher has a perfect estimate of each advertiser’s triggering action frequency
function, i.e. where Ũi (p) = U (p, qi ) for all i and p, the above mechanism is allocation and revenue
equivalent to a PPE mechanism with endogenous product prices.
The above solution is effective in a wide range of settings. It works well in settings where the
triggering action frequency is a stochastic function of product price and also in the important special
23
case of personalized ad settings where U (p, q) is a probability function. Its main practical drawback
is that it requires the advertiser to ex-ante commit to a product price and the publisher to be able
to monitor the extent to which the advertiser adheres to her commitment. It also complicates the
publisher’s learning task, as it requires him to have full knowledge of the triggering action frequency
function for all prices and all advertiser types.
6.2
History-based fee schedules
Section 4.2 shows that, in settings with repeated interaction, the publisher can induce the advertiser
to price her products at the efficient level by committing to “punish” her in the next round of
interaction if her current round triggering action frequency was not sufficiently high (or, in the
special case of personalized ads, if no triggering action was observed in the current round). This
punishment consists either of temporarily increasing the per-action fee charged to the advertiser or
of temporarily charging a fixed fee on top of the per-action fee. The advantage of such mechanisms
is that they do not require the advertiser to disclose product prices to the publisher.
In auction-based settings the publisher cannot directly control the per-action fee charged to
advertisers. Therefore, the most straightforward type of intervention would be to add a system of
fixed fees and subsidies on top of the per-action fee that is decided by auction. In the important case
of personalized ad settings I will prove that, for properly calibrated fees and subsidies, the following
two-state mechanism can induce efficient bidding and pricing:
Mechanism M2:
State H:
All advertisers begin the game at state H. While they are in state H advertisers pay an
auction participation fee fH (q). If they win the auction they are charged a fixed fee
gH (q) plus the per-action fee that is decided by auction. If a triggering action occurs
during the current period the advertiser remains in state H, otherwise she transitions to
state L.
State L:
For every period that advertisers are in state L they pay an auction participation fee
fL (q). If they win the auction they are charged a fixed fee gL (q) plus the per-action fee
that is decided by auction. If a triggering action occurrs during the current period the
advertiser transitions to state H otherwise she remains in state L.
The following proposition (analogous to Proposition 5) provides the details.
Proposition 12. Consider an infinite horizon RBR-PPA setting that is characterized by joint
advertiser-publisher profit V (p, q), triggering action probability U (p, q) and the above two-state mechanism M2 involving auction participation fees fH (q), fL (q) and fixed fees gH (q), gL (q). If the publisher has a correct assessment of every advertiser’s triggering action probability function U (p, q)
and sets the fees as follows:
24
fH (q) = 0
fL (q) =
gH (q) =
gL (q) =
1−δ
δG(q) Z(q)
(p∗ (q),q)
− 1−UG(q)
Z(q)
U (p∗ (q),q)
Z(q)
G(q)
where Z(q) is the expected per-action payment to the publisher, then the above mechanism:
1. induces all advertiser types to always price their products at p∗ (q) and to bid their expected
value-per-action W (p∗ (q), q)
2. results in identical allocation and expected lifetime discounted advertiser profits and publisher
revenue to those of an infinite horizon PPE mechanism with endogenous prices
Observe that the solution involves a positive auction participation fee in state L only. Furthermore,
notice that the fixed fee gH (q) charged to the winner at state H is negative, i.e. a subsidy.
The following is the intuition behind the above result and an explanation of why both auction
participation and fixed fees are needed. As was the case in Section 4.2 the threat of lower payoffs
(due to the participation fee fL (q)) in state L provides a disincentive to advertisers to raise the
price of their products too much; higher prices reduce the probability that a triggering action will
be observed in the current round and thus increase the probability of transitioning to (or remaining
in) state L. At the same time, the positive probability of transitioning to lower payoff state L if
one wins the auction and doesn’t generate a triggering action while at state H, reduces a state-H
advertiser’s expected valuation of winning the auction (and, thus, her expected bid). Subsidy gH (q)
is then necessary to correct for this distortion and restore the advertiser’s expected valuation of
winning the auction to her value-per-bid. Similarly, while in state L, the only way to transition
to the higher payoff state H is by winning the auction and generating a triggering action. This
increases a state-L advertiser’s expected valuation of winning the auction. Likewise, fixed fee gL (q)
is necessary to correct this distortion.
The above schedule of auction participation and fixed fees is calibrated so that their expected
lifetime discounted sum is zero. Therefore, they do not change either the advertisers’ or the publisher’s lifetime discounted payoffs relative to a mechanism (e.g. PPE) where advertisers bid and
price in identical ways but where there are no extra fees or subsidies involved.
Just as I did in Section 4.2 it is instructive to ponder whether one can induce efficient pricing by a “pure PPA” mechanism, i.e. without resorting to auction participation or fixed fees.
Lemma 1 suggests that it might be possible to construct effective punishment mechanisms by
exploiting an advertiser’s quality weight: the publisher can commit to increasing an advertiser’s
quality weight when triggering actions occur and decreasing it when they don’t. Although the
threat to decrease an advertiser’s quality weight might succeed in reducing, or eliminating, price
distortions, it suffers from another important drawback: to the extent that punishment periods
occur with positive probability it would also reduce the auction’s allocative efficiency. Recall that
RBR mechanisms allocate the slot to the bidder who achieves the maximum score si = ui bi , where
25
(by Proposition 8) bi = W (pA (qi ), qi ). Score si = ui W (pA (qi ), qi ) is perfectly correlated with
the advertiser’s value V (pA (qi ), qi ) = U (pA (qi ), qi )W (pA (qi ), qi ) and leads to allocative efficiency if
and only if ui = U (pA (qi ), qi ) for all i. If there are positive probability punishment states where
ui 6= U (pA (qi ), qi ) for some i then during such states the resource may not be allocated to the
advertiser who values it the most. This, in turn, reduces the mechanism’s expected lifetime joint
advertiser-publisher profit below the optimal. Positive probability punishment states occur in any
setting where the triggering action frequency is a stochastic function of product price or where
function U (p, q) denotes a probability. The following proposition ensues:
Proposition 13: In any setting where there is a stochastic relationship between the prices of
advertised products and the observed triggering action frequency/probability, the discounted lifetime
joint publisher-advertiser profit that can be achieved by a pure PPA mechanism where the advertiser’s
quality weight is a function of her triggering action frequency history is strictly lower than what can
be achieved by Mechanism M2.
In conclusion, restoring efficient product pricing in search advertising auctions requires either the
ability to credibly elicit the advertiser’s current round product price or payment mechanisms where
the publisher charges auction participation fees and fixed fees/subsidies on top of the performancebased fees.
7
Concluding Remarks
Technological advances have made it increasingly feasible to track the impact of individual advertising messages on consumer behavior. Accordingly, pay-per-performance advertising mechanisms,
whereby the publisher is only paid when consumers perform certain actions (e.g. clicks, calls,
purchases) that are tied to a specific advertising stimulus, have been gaining ground. Such pay-peraction (PPA) mechanisms are proving popular with advertisers because they help limit their risk
when investing in new and often untested advertising technologies as well as allow them to better
estimate their advertising ROI.
This paper highlights an important, and previously unnoticed, side-effect of PPA advertising.
I show that, in settings where at least a subset of consumers has some information about product
price before performing the action (click, call, purchase, etc.) that triggers payment to the publisher,
PPA mechanisms induce advertisers to distort the prices of their products - usually upwards - as it
is more beneficial to them to pay the publisher fewer times but realize a higher net profit per sale.
Such equilibria always reduce social welfare and reduce the payoffs of most stakeholders involved:
consumers are always left with a lower surplus (because they pay higher prices) and one or both of
advertiser profits and publisher revenues decline.
Price distortions persist in the rank-by-revenue auction-based variants of PPA advertising currently practiced by Google and Yahoo. Interestingly, in the latter settings they always reduce
publisher revenues relative to more traditional pay-per-exposure methods.
Fortunately, it is possible to introduce relatively simple enhancements to these mechanisms that
26
restore efficient pricing. I propose two broad sets of ideas. The first idea requires the advertiser
to disclose to the publisher both her bid as well as the product price she intends to charge. The
publisher then makes the advertiser’s current round quality weight a function of both her past
history and her current round product price. The second idea is to construct mechanisms whereby
advertisers are charged some form of penalty, e.g. an auction participation fee, following periods
where no triggering actions (i.e. clicks, sales, etc.) occur. The resulting mechanisms maximize the
joint publisher-advertiser profit and have identical revenue and allocation properties to an optimal
pay-per-exposure mechanism.
To keep my models tractable but also to better highlight the phenomena that form the focus of
the paper I made a number of simplifying assumptions. I am arguing that these assumptions do not
detract from the essence of the phenomenon.
First, I assumed that both the publisher and the advertisers are risk neutral. This assumption
was important to retain tractability. Real-life advertisers tend to be risk-averse; in fact, risk aversion
is an important motivator for the introduction of pay-per-performance payment methods (Mahdian
and Tomak 2008). However, although risk aversion will introduce some additional forces in favor
of performance-based advertising, the fundamental drivers of price distortion, that are due to the
publisher and advertisers each independently attempting to maximize their payoffs, will persist.
Second, I have assumed that the advertising resource is the only channel through which advertisers can connect to consumers or alternatively, that if the advertiser also sells her products through
additional channels, she sets her prices separately for each channel. If this assumption is not true,
for example if the advertiser sells through multiple channels and charges the same prices across all
channels, price distortions will still occur, however their magnitude will be smaller and proportional
to the relative share of revenues that can be attributed to the advertising resource.
Third, I assumed that the publisher is a monopolist. As argued in Section 3 this is not an
essential assumption: as long as the publisher has some market power (i.e. if he can charge the
advertisers a per-action fee above marginal cost) the price distortions discussed in this paper will
occur.
Internet technologies have spurred a tremendous wave of innovation in advertising methods.
New ideas are being continuously invented and tried out by ambitious entrepreneurs, often without
being rigorously analyzed. Given the fast pace of competition and innovation in the Internet arena
it is only natural that some of these ideas might have shortcomings or side-effects that are not
immediately obvious to their inventors. One role for academia in this fast-changing environment is
to place these new ideas under a rigorous theoretical lens, identify their shortcomings and propose
economically sound improvements. This work is very much in this spirit.
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30
Proofs
Proof of Lemma 1
It is:
ΠA (q, u; bA (q, u), pA (q, u)) = V (pA (q, u), q)Ψ(ubA (q, u)) − U (pA (q, u), q)Z A (bA (q, u), u)
8)
bA (q, u)
=
´ ub
s 0
0 u Ψ (s)ds. Differentiating with respect to u and substituting
W (pA (q, u), q) = V (pA (q, u), q)/U (pA (q, u), q) we obtain:
where Z A (b, u) =
(see Proposition
Z A (W (pA (q, u), q), u)
∂ΠA (q, u)
=
>0
∂u
u
Proof of Proposition 1
The proof is sketched in the main body of the text.
Proof of Proposition 2
The advertiser’s profit function is equal to πr (p|w) = U (p)(W (p) − w) = V (p) − U (p)w. If w =
f
U (p)
the profit function becomes πr (p|w) = V (p) − f . The price pA that maximizes πr (p|w) is then equal
to the price p∗ that maximizes V (p). The constant f must be less than V (p∗ ) − u where u is the
advertiser’s reservation utility.
Proof of Proposition 3
If each period’s observed triggering action frequency is a deterministic function of product price it
is Ut = U (pt ). The Bellman equation of the infinite horizon PPA setting of interest to this context
then is:
Ω(wt ) = max [V (p) − U (p)wt + δΩ(h(U (p)))]
p
where
·
µ
¶¸
1
U
h(U ) = w0 1 +
1−
δ
U (p∗ )
and p∗ is the price that maximizes V (p), i.e. satisfies V 0 (p∗ ) = 0 and V 00 (p∗ ) < 0.
Notice that in the above system product prices pt = p(wt ) only depend on the current period’s
per-action fee wt . Specifically, they must satisfy the first order condition:
£
¤
V 0 (p(wt )) − U 0 (p(wt )) wt − δΩ0 (h(U (p(wt ))))h0 (U (p(wt ))) = 0
and the corresponding second order condition. Substituting:
Ω0 (wt ) = −U (p(wt ))
31
(8)
h0 (U ) = −
w0
δU (p∗ )
into (8) we obtain:
·
¸
U (p(h(U (p(wt )))))
V (p(wt )) − U (p(wt )) wt − w0
=0
U (p∗ )
0
0
(9)
I will show that, for wt = w0 , p(w0 ) = p∗ solves (9). Observe that it is h(U (p∗ )) = w0 . Substituting
wt = w0 and p(wt ) = p(w0 ) = p∗ into (9) we obtain:
·
¸
U (p∗ )
= V 0 (p∗ ) = 0
V (p ) − U (p ) w0 − w0
U (p∗ )
0
∗
0
∗
For wt = w0 and p(w0 ) = p∗ the second-order condition that corresponds to (9) is:
∗
00 ∗
0
∗
0
∗
V 00 (p
h ) − U (p ) [w0 − δΩ (h(U (p )))h (U (p ))]
i
+δ Ω00 (h(U (p∗ )) [h0 (U (p∗ ))U 0 (p∗ )]2 + Ω0 (h(U (p∗ )))h00 (U (p∗ )) [U 0 (p∗ )]2 < 0
Because it is Ω00 (w) = 0, h00 (U ) = 0 and w0 − δΩ0 (h(U (p∗ )))h0 (U (p∗ )) = 0 the above simplifies to
V 00 (p∗ ) < 0 which is true by definition.
In summary, I have shown that if the current period per-action fee is w0 and the next period
fee is equal to h(U ), the current period product price that maximizes the advertiser’s value function
is equal to p∗ . Furthermore (because it is h(U (p∗ )) = w0 ) if wt = w0 then it is also wt+1 = w0 .
Therefore, at equilibrium the publisher will charge the advertiser a per-period fee equal to w0 at all
periods and the advertiser will find it optimal to charge a product price p(w0 ) = p∗ at all periods.
Proof of Proposition 4
The Bellman equations that describe the system of interest in this proposition are:
ΩH = V (pH ) − U (pH )wH + δ [U (pH )ΩH + (1 − U (pH ))ΩL ]
ΩL = −f + V (pL ) − U (pL )wL + δ [U (pL )ΩH + (1 − U (pL ))ΩL ]
(10)
where pH , pL satisfy the first-order conditions:
V 0 (pH ) + U 0 (pH ) [−wH + δ (ΩH − ΩL )] = 0
V 0 (pL ) + U 0 (pL ) [−wL + δ (ΩH − ΩL )] = 0
The requirement pH = pL = p∗ , such that V (p∗ ) = 0 implies:
wH = wL = δ (ΩH − ΩL )
Substituting (13) into (14) and solving for f, wH , wL under the constraint ΩH =
(11)
u
1−δ
that character-
izes the situation where the publisher extracts all the surplus subject to leaving the advertiser with
32
an average per-round reservation utility u, gives:
f = (V (p∗ ) − u) 1δ
wH = wL = (V (p∗ ) − u)
Proof of Proposition 5
The Bellman equations that describe the system of interest in this proposition are:
ΩH = s + V (pH ) − U (pH )wH + δ [U (pH )ΩH + (1 − U (pH ))ΩL ]
ΩL = −f + V (pL ) − U (pL )wL + δ [U (pL )ΩH + (1 − U (pL ))ΩL ]
(12)
where pH , pL satisfy the first-order conditions:
V 0 (pH ) + U 0 (pH ) [−wH + δ (ΩH − ΩL )] = 0
V 0 (pL ) + U 0 (pL ) [−wL + δ (ΩH − ΩL )] = 0
The requirement pH = pL = p∗ , such that V (p∗ ) = 0 implies:
wH = wL = δ (ΩH − ΩL )
(13)
Let NH denote the net present value of the system of non-performance-based fees f and subsidies
s. It is:
NH = s + δ [U (p∗ )NH + (1 − U (p∗ ))NL ]
NL = −f + δ [U (p∗ )NH + (1 − U (p∗ ))NL ]
(14)
Solving the above system and setting NH = 0 gives:
s=f
δ(1 − U (p∗ ))
1 − δ(1 − U (p∗ ))
(15)
Substituting (13) and (15) into (12) and solving for f, wH , wL under the constraint ΩH =
u
1−δ
that characterizes the situation where the publisher extracts all the surplus subject to leaving the
advertiser with an average per-round reservation utility u, gives:
∗
(p )
s = (V (p∗ ) − u) 1−U
U (p∗ )
∗ ))
(p
f = (V (p∗ ) − u) 1−δ(1−U
δU (p∗ )
wH = wL =
V (p∗ )−u
U (p∗ )
Proof of Proposition 6
Consider a setting where each period the publisher charges the advertiser a per-action fee w(h)
where h is the advertiser’s history of past triggering actions. The Bellman equations that describe
33
the system of interest in this proposition are:
Ω(h) = V (p(h)) − U (p(h))w(h) + δ [U (p(h))Ω(h+ (h)) + (1 − U (p(h)))Ω(h− (h))]
(16)
where h+ (h), h− (h) denote the history following the occurrence (non-occurrence) of a triggering
action in the current round respectively and where p(h) satisfies the first-order condition:
V 0 (p(h)) + U 0 (p(h)) [−w(h) + δ (Ω(h+ (h)) − Ω(h− (h)))] = 0
The requirement p(h) = p∗ , such that V (p∗ ) = 0 implies:
w(h) = δ (Ω(h+ (h)) − Ω(h− (h)))
for all h
(17)
Substituting (17) into (16) gives:
Ω(h) = V (p∗ ) + δΩ(h− (h))
for all h
(18)
If there is no termination state, i.e. no state for which equation (18) does not hold then the above
equation has the solution Ω(h) =
V (p∗ )
1−δ
for all h, which implies that w(h) = 0 for all h.
A termination state is a state where the game would end for the advertiser. For example a state
that, once reached, the advertiser would no longer be allowed to obtain the resource. Such a state
would have zero payoff. Although such a state can be potentially reachable through multiple paths,
repeated application of equation (18) shows that such a state is always reached from an initial state
h0 after k consecutive periods where no triggering action occurs, for some integer k ≥ 1. Then, by
recursive application of (18) the advertiser’s lifetime payoff is equal to:
Ω(h0 ) = V (p∗ ) + δΩ(h− (h0 )) = V (p∗ ) + δ [V (p∗ ) + δΩ(h− (h− (h0 )))]
= (1 + δ + ... + δ k−1 )V (p∗ ) =
1−δ k
1−δ V
(p∗ )
The probability that there will be k consecutive periods without a triggering action is τ =
(1 − U (p∗ ))k . Therefore, the probability that an advertiser will be expelled at or before state i + k
is given by the CDF of a geometric distribution with parameter τ . For any probability P arbitrarily
close to 1 there is, thus, a finite i, such that the probability that the advertiser will be expelled at or
before state i+k is higher than P . This means that the discounted lifetime joint advertiser-publisher
profit in this mechanism is always lower than that achievable by an infinite horizon mechanism where
the per-period joint profit was always equal to the maximum V (p∗ ).
Proof of Proposition 7
Parts 1 and 2
£ ¤0
Assume that every other bidder bids β E (y) and that (as I will show) β E (y) > 0. An advertiser
of type q will choose bid bE (q) and price pE (q) that maximize:
34
−1
ΠE (q; bE (q), pE (q), β E (·)) =
[β E ] ˆ (bE (q))
¡
¢
V (pE (q), q) − β E (y) G0 (y)dy
q
First-order conditions with respect to bid and price give:
£ ¤−1 E
¢
£ ¤−1 E
£ ¤−1 E
∂ βE
(b (q)) ¡
V (pE (q), q) − bE (q) G0 ( β E
(b (q))) = 0 and V1 (pE (q), q)G( β E
(b (q))) = 0
E
∂b (q)
£ ¤−1 E
At a symmetric equilibrium it must be bE (q) = β E (q) which implies that G( β E
(b (q))) =
E −1 (bE (q))
£
¤
∂
β
−1
[
]
= [bE 1]0 (q) > 0. The above then reduces
G(q) > 0, G0 ( β E
(bE (q))) = G0 (q) > 0 and
∂bE (q)
to:
bE (q) = V (pE (q), q)
and
V1 (pE (q), q) = 0
Assumption A1 implies that pE (q) = p∗ (q) is uniquely defined for all q and also that V11 (pE (q), q) <
∗
0. Assumption A3 and the envelope theorem further imply that ∂V (p∂q(q),q) > 0 and hence that
£ E ¤0
b (q) > 0, as originally assumed. The corresponding Hessian matrix is:
"
0
H E (bE (q), pE (q), q) =
− V2 (pGE(q)
(q),q)
0
0
V11 (pE (q), q)G(q)
#
It is straightforward to show that H E is negative definite and, therefore, that the above pair
(bE (q), pE (q)) corresponds to a local maximum of ΠE (q; bE (q), pE (q), β E (·)) for all q.
Part 3
In a PPE setting where prices are exogenously set standard auction theory predicts that each bidder
will bid her expected valuation bE (q; p(·)) = V (p(q), q). Substituting into (2):
ˆ
E
q
È
R (p(·)) = N
q
z
!
0
V (p(y), y)G (y)dy F 0 (z)dz
q
Because V (p∗ (y), y) ≥ V (p(y), y) for all y (with equality iff p∗ (y) = p(y)), it is RE = RE (p∗ (·)) ≥
RE (p(·)) with equality iff p∗ (·) = p(·).
Proof of Proposition 8
Part 1
An advertiser of type q will choose bid bA (q, u) and price pA (q, u) that maximize (3). The latter can
be equivalently rewritten as:
35
ΠA (q, u; bA (q, u), pA (q, u)) = V (pA (q, u), q)Ψ(ubA (q, u)) − U (pA (q, u), q)Z A (bA (q, u), u)
where Z A (b, u) =
´ ub
0
s 0
u Ψ (s)ds.
Differentiating with respect to pA (q, u) gives:
∂ΠA
= V1 (pA (q, u), q)Ψ(ubA (q, u)) − U1 (pA (q, u), q)Z A (bA (q, u), u)
∂pA (q, u)
(19)
Assumption A1 implies that:
V1 (p, q) > 0 for all p < p∗ (q)
V1 (p, q) = 0 for p = p∗ (q)
V1 (p, q) < 0 for all p >
(20)
p∗ (q)
• If U1 (p, q) < 0 for all p then (19) and (20) imply that
∂ΠA
∂pA (q,u)
> 0 for all pA (q, u) ≤ p∗ (q)
and, therefore, that the advertiser can strictly increase net profits if she raises the price of her
products above p∗ (q) .
• If U1 (p, q) > 0 for all p then (19) and (20) imply that
∂ΠA
∂(−pA (q,u))
∂ΠA
∂pA (q,u)
< 0 (which is equivalent to
> 0) for all pA (q, u) ≥ p∗ (q) and, therefore, that the advertiser can strictly increase
net profits if she reduces the price of her products below p∗ (q) .
• Finally, if U1 (p, q) = 0 for all p then (19) and (20) imply that
∂ΠA
∂pA (q,u)
= V1 (pA (q, u), q)Ψ(ubA (q, u))
and, therefore, that the price that maximizes V (·) also maximizes ΠA (·)
Note that the above hold for all u and for any positive bA (q, u).
Part 2
For arbitrary u the first-order condition with respect to bid gives:
¡
¢
U (pA (q, u), q) W (pA (q, u), q) − bA (q, u) Ψ0 (ubA (q, u))u = 0
which implies:
bA (q, u) = W (pA (q, u), q)
Part 3
In the special case where u = U (pA (q), q), the first-order condition with respect to bid gives:
¡
¢
U (pA (q), q) W (pA (q), q) − bA (q) Ψ0 (U (pA (q), q)bA (q)) = 0
which implies:
bA (q) = W (pA (q), q)
36
Taking the first order condition with respect to price and substituting the expression for bA (q) we
obtain:
V1 (pA (q), q)Ψ(U (pA (q), q)W (pA (q), q)) − U1 (pA (q), q)Z A (W (pA (q), q), U (pA (q), q)) = 0
(21)
Note that in the special setting where (a) everyone bids their expected value-per-action. and (b)
quality weights are equal to everyone’s triggering action frequency, every advertiser’s score s = ub
is simply equal to her expected value function V (pA (q), q). If we assume that
∂
∂q V
(pA (q), q) ≥ 0,
i.e. that equilibrium value functions are monotonically increasing with q then advertiser scores will
also be monotonically increasing with q. Under these assumptions the distribution Ψ(s) of everyone
else’s scores is identical to the distribution G(q) of everyone else’s type, where s = V (pA (q), q).
Substituting into (21) we obtain:
V1 (pA (q), q)G(q) − U1 (pA (q), q)Z A (q) = 0
where
´ U (pA (q),q)W (pA (q),q) 0
sΨ (s)ds
Z A (q) = Z A (W (pA (q), q), U (pA (q), q)) = U (pA1(q),q) 0
´ V (pA (q),q) 0
´q
1
1
A
0
= U (pA (q),q) 0
sΨ (s)ds = U (pA (q),q) q V (p (y), y)G (y)dy
Proof of Proposition 9
Part 1
Substituting u = U (pA (q), q) and bA (q) = W (pA (q), q) into (3) I obtain:
³
´
´ U (pA (q),q)W (pA (q),q)
U (pA (q), q) W (pA (q), q) − U (pAs(q),q) Ψ0 (s)ds
ΠA (q, U (pA (q), q); bA (q), pA (q)) = 0
¢
¢
´q¡
´ V (pA (q),q) ¡
= 0
V (pA (q), q) − s Ψ0 (s)ds = q V (pA (q), q) − V (pA (y), y) G0 (y)dy
´q
= V (pA (q), q)G(q) − q V (pA (y), y)G0 (y)dy = ΠE (pA (·))
Part 2
Substituting u = U (pA (q), q) and bA (q) = W (pA (q), q) into (4) I obtain:
³´ A
´q
U (p (q),q)W (pA (q),q)
RA (bA (·, ·), pA (·, ·)) = N q U (pA (q), q) 0
´
´ q ³´ V (pA (q),q) 0
=N q
sΨ
(s)ds
F 0 (q)dq
0
´
³
´q ´q
A (y), y)G0 (y)dy F 0 (q)dq = R (pA (·))
V
(p
=N q
E
q
37
s
U (pA (q),q)
´
Ψ0 (s)ds F 0 (q)dq
Proof of Proposition 10
Part 1
From Proposition 8, at an RBR-PPA equilibrium it is:
ΠA (q; pA (q), ζ A (q)) = V (pA (q), q)G(q) − U (pA (q), q)Z A (q) = V (pA (q), q)G(q)(1 − ζ A (q))
(22)
From Proposition 8 we also know that, if U1 (p, q) < 0, it is pA (q) > p∗ (q) , which, by Assumption
A1 implies that V1 (pA (q), q) < 0. From monotone comparative statics theory (Milgrom and Shannon 1994) a sufficient condition for ∂pA (q)/ζ A (q) > 0 is that ∂ 2 ΠA (q; p(q), ζ(q))/∂p(q)∂ζ(q) > 0.
Differentiating (22) I obtain:
∂ 2 ΠA (q; pA (q), ζ A (q))
= −V1 (pA (q), q)G(q) > 0
∂pA (q)∂ζ A (q)
Part 2
The price pA (q) that maximizes ΠA (q; p(q), ζ(q)) must satisfy the first-order condition:
V1 (pA (q), q)G(q) − U1 (pA (q), q)Z A (q)
¡
¢
= U1 (pA (q), q) W (pA (q), q)G(q) − Z A (q) + U (pA (q), q)W1 (pA (q), q)G(q) = 0
(23)
At the limit where ζ A (q) → 1 it is W (pA (q), q)G(q) − Z A (q) → 0 (follows directly from the definition
ζ A (q) =
Z A (q)
).
W (pA (q),q)G(q)
Then (23) simplifies to:
U (pA (q), q)W1 (pA (q), q)G(q) = 0
If W1 (p, q) > 0 the above implies U (pA (q), q) = 0, which also implies V (pA (q), q) = 0.
Proof of Proposition 11
An advertiser of type q will choose bid bA (q, u) and price pA (q, u) that maximize (3). The latter can
be equivalently rewritten as:
ΠA (q, u; bA (q, u), pA (q, u)) = V (pA (q, u), q)Ψ(ubA (q, u)) − U (pA (q, u), q)Z A (bA (q, u), u)
with respect to
´ ub
s 0
u Ψ (s)ds.
bA (q), pA (q) give:
where Z A (b, u) =
0
In the special case where u = U (pA (q), q) first-order conditions
£
¤
Ψ0 (U (pA (q), q)bA (q))U (pA (q), q) V (pA (q), q) − U (pA (q), q)bA (q) = 0
V1 (pA (q), q)Ψ(U (pA (q), q)bA (q))
£
¤
+Ψ0 (U (pA (q), q)bA (q))U1 (pA (q), q)bA (q) V (pA (q), q) − U (pA (q), q)bA (q) = 0
38
Solving we obtain:
bA (q) =
V (pA (q),q)
U (pA (q),q)
= W (pA (q), q)
V1 (pA (q), q) = 0
which in turn implies that pA (q) = p∗ (q) and, by Proposition 9, that the mechanism is allocation
and revenue equivalent to a PPE mechanism with endogenous product prices.
Proof of Proposition 12
The Bellman equations that describe the system of interest in this proposition are:
ΩH (q, u) = −fH (q) + (−gH (q) + V (pH , q))Ψ(ubH (q)) − U (pH (q), q)Z(u, bH (q))
+δΨ(ubH (q)) [U (pH (q), q)ΩH (q, u) + (1 − U (pH (q), q))ΩL (q, u)] + δ(1 − Ψ(ubH (q)))ΩH (q, u)
ΩL (q, u) = −fL (q) + (−gL (q) + V (pL (q), q))Ψ(ubL (q)) − U (pL (q), q)Z(u, bH (q))
+δΨ(ubL (q)) [U (pL (q), q)ΩH (q, u) + (1 − U (pL (q), q))ΩL (q, u)] + δ(1 − Ψ(ubL (q)))ΩL (q, u)
(24)
where
1
Z(u, b) =
u
ˆ
ub
sΨ0 (s)ds
0
and bH (q), bL (q), pH (q), pL (q) satisfy the first-order conditions:
−gH (q) + U (pH (q), q) [W (pH (q), q) − bH (q)] − δ(1 − U (pH (q), q))(ΩH (q, u) − ΩL (q, u)) = 0
−gL (q) + U (pL (q), q) [W (pL (q), q) − bL (q)] + δU (pH (q), q)(ΩH (q, u) − ΩL (q, u)) = 0
V1 (pH (q), q)Ψ(ubH (q)) + U1 (pH (q), q) [−Z(u, bH (q)) + δΨ(ubH (q))(ΩH (q, u) − ΩL (q, u))] = 0
V1 (pL (q), q)Ψ(ubL (q)) + U1 (pL (q), q) [−Z(u, bL (q)) + δΨ(ubL (q))(ΩH (q, u) − ΩL (q, u))] = 0
We would like to set the schedule of fees and subsidies so that (for all q) advertisers (i) price
their products at the socially efficient level p∗ (q) defined by V1 (p∗ (q), q) = 0, (ii) bid their expected
value-per-action W (p∗ (q), q) and (iii) obtain a lifetime discounted payoff equal to the payoff they
would obtain if they bid and priced that way without any fees or subsidies imposed. Mathematically
these requirements are equivalent to:
pH (q) = pL (q) = p∗ (q)
bH (q) = bL (q) = W (p∗ (q), q)
ΩH (q, u) =
V (p∗, (q),q)Ψ(ubH (q))−U (p∗ (q),q)Z(u,bH (q))
1−δ
and, in turn, imply the following:
39
gH (q) = −δ(1 − U (p∗ (q), q))(ΩH (q, u) − ΩL (q, u))
gL (q) = δU (p∗ (q), q)(ΩH (q, u) − ΩL (q, u))
(25)
Z(u, bH (q)) = δΨ(ubH (q))(ΩH (q, u) − ΩL (q, u))
ΩH (q, u) =
V (p∗, (q),q)Ψ(ubH (q))−U (p∗ (q),q)Z(u,bH (q))
1−δ
Substituting (24) into (25) and solving for fH (q), fL (q), gH (q), gL (q) we obtain:
fH (q) = 0
fL (q) =
gH (q) =
gL (q) =
where Z(q) = Z(u, W (p∗ (q), q)) =
1
u
1−δ
δΨ(uW (p∗ (q),q)) Z(q)
1−U (p∗ (q),q)
− Ψ(uW
(p∗ (q),q)) Z(q)
U (p∗ (q),q)
Ψ(uW (p∗ (q),q)) Z(q)
´ uW (p∗ (q),q)
0
sΨ0 (s)ds. If the publisher sets each advertiser’s
quality weight equal to her triggering action frequency and if, additionally, it is
∂
∂q V
(pA (q), q) ≥ 0
for all q then (see proof of Proposition 8) it is Ψ(uW (p∗ (q), q)) = Ψ(U (p∗ (q), q)W (p∗ (q), q)) =
Ψ(V (p∗ (q), q)) = G(q) and the above simplify to:
fH (q) = 0
fL (q) =
gH (q) =
gL (q) =
1−δ
δG(q) Z(q)
(p∗ (q),q)
− 1−UG(q)
Z(q)
U (p∗ (q),q)
Z(q)
G(q)
Furthermore, under the above assumptions, from Proposition 9 the numerator of ΩH (q) is identical
to the advertiser’s payoff in a PPE mechanism with optimal endogenous pricing. Therefore, the
advertiser’s lifetime discounted payoff is identical to the lifetime discounted payoff of an infinite
sequence of PPE mechanisms with endogenous prices. It is similarly easy to show that the net
lifetime discounted sum of fees fH (q), fL (q), gH (q), gL (q) is equal to zero. Therefore, the publisher’s
revenue is also identical to that of an infinite sequence of PPE mechanisms with endogenous prices.
Proof of Proposition 13
The proof is sketched in the main body of the text.
Appendix: Example of a pure PPA mechanism that induces efficient
product pricing
Consider a mechanism that has the following properties:
1. All advertisers begin the game at an initial “perfect” state 0.
2. During a period where the advertiser’s state is i the publisher charges a per-action fee wi
40
3. At the end of a period where the advertiser’s state is i: If a triggering action occurs, the
advertiser transitions back to state 0. If no triggering action occurs the advertiser transitions
to state i + 1
4. Once the advertiser reaches state k she is expelled from the game (i.e. no longer allowed to
obtain the advertising resource)
It is easy to see that the above mechanism is a special case of the general class of mechanisms
described by Proposition 6. Therefore, for an appropriately chosen schedule of fees wi (i = 0, 1, .., k−
1), it can induce advertisers to always price their products at pA = p∗ . In this appendix I provide
a full analysis of the mechanism.
Specializing the general results of Proposition 6 to this special case we immediately obtain the
following:
1. The Bellman equations that describe the advertiser’s payoffs are:
(
Ωi =
V (p∗ ) − U (p∗ )wi + δ [U (p∗ )Ω0 + (1 − U (p∗ ))Ωi+1 ] =
1−δ k−i
1−δ V
(p∗ )
0
i<k
i=k
2. The publisher’s fee schedule is given by:
wi = δ (Ω0 − Ωi+1 ) = δ
δ k−i−1 − δ k
V (p∗ ) i = 0, 1, ..., k − 1
1−δ
The lifetime discounted joint advertiser-publisher profit is J0 , where J0 is the solution of:
(
Ji =
V (p∗ ) + δ [U (p∗ )J0 + (1 − U (p∗ ))Ji+1 ]
i<k
0
i=k
Taking successive differences:
(
Ji − Ji+1 =
δ(1 − U (p∗ )) (Ji+1 − Ji+2 )
V
(p∗ )
+
δU (p∗ )J0
i<k−1
i=k−1
and backwards-substituting into J0 we obtain:
J0 =
1 − δ k (1 − U (p∗ ))k
V (p∗ )
1 − δ[1 − δ k (1 − U (p∗ ))k U (p∗ )]
The publisher’s discounted lifetime profit is simply:
·
Ri = J0 − Ω0 =
¸
1 − δ k (1 − U (p∗ ))k
1 − δk
V (p∗ )
−
1−δ
1 − δ[1 − δ k (1 − U (p∗ ))k U (p∗ )]
It is easy to see that:
1. Both Ω0 and J0 are monotonically increasing functions of k.
41
Figure 2: Publisher’s revenue R0 as a function of the threshold k of consecutive rounds without
triggering actions after which an advertiser is expelled.
2. At the limit k → ∞ , Ω0 → J0 →
1
1−δ V
(p∗ ), wi → 0 for all i, and R0 → 0
For finite values of k, as k increases R0 increases, then decreases. There is, therefore, an optimal
value k ∗ that maximizes the publisher’s profits. The intuition behind this result is the following:
When k is small (i.e. when advertisers get expelled from the game after a relatively small number
of consecutive rounds without an observed triggering action), advertisers leave the game too soon,
which hurts the publisher’s lifetime profit per advertiser. When k is large then most of the time the
fees charged by the publisher are very low, which again hurts his lifetime profit per advertiser.
Figure 2 plots R0 for δ = 0.9, V (p∗ ) = 1 and U (p∗ ) = 1/2. The integer value that maximizes R0
is k ∗ = 5. Setting k = 5 to the above formulae we obtain:
J0 = 9.06
w0 = 0.59
Ω0 = 4.09
w1 = 1.25
R0 = 4.97
w2 = 1.97
w3 = 2.78
w4 = 3.68
Notice that the split of the joint profits J0 between the publisher (R0 ) and the advertiser (Ω0 ) is
relatively equitable (55% goes to the publisher and 45% goes to the advertiser). Also, the per-action
publisher fees wi monotonically increase as the number of consecutive periods with no triggering
action increases. Finally, per Proposition 6, the discounted lifetime joint advertiser-publisher profit
J0 = 9.06 that is attainable with this mechanism is strictly below the optimal J ∗ =
V (p∗ )
1−δ =10
that
is attainable through the mechanisms analyzed in Propositions 4 and 5. The latter mechanisms
involve the use of both performance-based and non-performance-based fees.
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